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G = C3×C105order 315 = 32·5·7

Abelian group of type [3,105]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C105, SmallGroup(315,4)

Series: Derived Chief Lower central Upper central

C1 — C3×C105
C1C7C35C105 — C3×C105
C1 — C3×C105
C1 — C3×C105

Generators and relations for C3×C105
 G = < a,b | a3=b105=1, ab=ba >


Smallest permutation representation of C3×C105
Regular action on 315 points
Generators in S315
(1 218 183)(2 219 184)(3 220 185)(4 221 186)(5 222 187)(6 223 188)(7 224 189)(8 225 190)(9 226 191)(10 227 192)(11 228 193)(12 229 194)(13 230 195)(14 231 196)(15 232 197)(16 233 198)(17 234 199)(18 235 200)(19 236 201)(20 237 202)(21 238 203)(22 239 204)(23 240 205)(24 241 206)(25 242 207)(26 243 208)(27 244 209)(28 245 210)(29 246 106)(30 247 107)(31 248 108)(32 249 109)(33 250 110)(34 251 111)(35 252 112)(36 253 113)(37 254 114)(38 255 115)(39 256 116)(40 257 117)(41 258 118)(42 259 119)(43 260 120)(44 261 121)(45 262 122)(46 263 123)(47 264 124)(48 265 125)(49 266 126)(50 267 127)(51 268 128)(52 269 129)(53 270 130)(54 271 131)(55 272 132)(56 273 133)(57 274 134)(58 275 135)(59 276 136)(60 277 137)(61 278 138)(62 279 139)(63 280 140)(64 281 141)(65 282 142)(66 283 143)(67 284 144)(68 285 145)(69 286 146)(70 287 147)(71 288 148)(72 289 149)(73 290 150)(74 291 151)(75 292 152)(76 293 153)(77 294 154)(78 295 155)(79 296 156)(80 297 157)(81 298 158)(82 299 159)(83 300 160)(84 301 161)(85 302 162)(86 303 163)(87 304 164)(88 305 165)(89 306 166)(90 307 167)(91 308 168)(92 309 169)(93 310 170)(94 311 171)(95 312 172)(96 313 173)(97 314 174)(98 315 175)(99 211 176)(100 212 177)(101 213 178)(102 214 179)(103 215 180)(104 216 181)(105 217 182)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105)(106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210)(211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315)

G:=sub<Sym(315)| (1,218,183)(2,219,184)(3,220,185)(4,221,186)(5,222,187)(6,223,188)(7,224,189)(8,225,190)(9,226,191)(10,227,192)(11,228,193)(12,229,194)(13,230,195)(14,231,196)(15,232,197)(16,233,198)(17,234,199)(18,235,200)(19,236,201)(20,237,202)(21,238,203)(22,239,204)(23,240,205)(24,241,206)(25,242,207)(26,243,208)(27,244,209)(28,245,210)(29,246,106)(30,247,107)(31,248,108)(32,249,109)(33,250,110)(34,251,111)(35,252,112)(36,253,113)(37,254,114)(38,255,115)(39,256,116)(40,257,117)(41,258,118)(42,259,119)(43,260,120)(44,261,121)(45,262,122)(46,263,123)(47,264,124)(48,265,125)(49,266,126)(50,267,127)(51,268,128)(52,269,129)(53,270,130)(54,271,131)(55,272,132)(56,273,133)(57,274,134)(58,275,135)(59,276,136)(60,277,137)(61,278,138)(62,279,139)(63,280,140)(64,281,141)(65,282,142)(66,283,143)(67,284,144)(68,285,145)(69,286,146)(70,287,147)(71,288,148)(72,289,149)(73,290,150)(74,291,151)(75,292,152)(76,293,153)(77,294,154)(78,295,155)(79,296,156)(80,297,157)(81,298,158)(82,299,159)(83,300,160)(84,301,161)(85,302,162)(86,303,163)(87,304,164)(88,305,165)(89,306,166)(90,307,167)(91,308,168)(92,309,169)(93,310,170)(94,311,171)(95,312,172)(96,313,173)(97,314,174)(98,315,175)(99,211,176)(100,212,177)(101,213,178)(102,214,179)(103,215,180)(104,216,181)(105,217,182), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315)>;

G:=Group( (1,218,183)(2,219,184)(3,220,185)(4,221,186)(5,222,187)(6,223,188)(7,224,189)(8,225,190)(9,226,191)(10,227,192)(11,228,193)(12,229,194)(13,230,195)(14,231,196)(15,232,197)(16,233,198)(17,234,199)(18,235,200)(19,236,201)(20,237,202)(21,238,203)(22,239,204)(23,240,205)(24,241,206)(25,242,207)(26,243,208)(27,244,209)(28,245,210)(29,246,106)(30,247,107)(31,248,108)(32,249,109)(33,250,110)(34,251,111)(35,252,112)(36,253,113)(37,254,114)(38,255,115)(39,256,116)(40,257,117)(41,258,118)(42,259,119)(43,260,120)(44,261,121)(45,262,122)(46,263,123)(47,264,124)(48,265,125)(49,266,126)(50,267,127)(51,268,128)(52,269,129)(53,270,130)(54,271,131)(55,272,132)(56,273,133)(57,274,134)(58,275,135)(59,276,136)(60,277,137)(61,278,138)(62,279,139)(63,280,140)(64,281,141)(65,282,142)(66,283,143)(67,284,144)(68,285,145)(69,286,146)(70,287,147)(71,288,148)(72,289,149)(73,290,150)(74,291,151)(75,292,152)(76,293,153)(77,294,154)(78,295,155)(79,296,156)(80,297,157)(81,298,158)(82,299,159)(83,300,160)(84,301,161)(85,302,162)(86,303,163)(87,304,164)(88,305,165)(89,306,166)(90,307,167)(91,308,168)(92,309,169)(93,310,170)(94,311,171)(95,312,172)(96,313,173)(97,314,174)(98,315,175)(99,211,176)(100,212,177)(101,213,178)(102,214,179)(103,215,180)(104,216,181)(105,217,182), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315) );

G=PermutationGroup([(1,218,183),(2,219,184),(3,220,185),(4,221,186),(5,222,187),(6,223,188),(7,224,189),(8,225,190),(9,226,191),(10,227,192),(11,228,193),(12,229,194),(13,230,195),(14,231,196),(15,232,197),(16,233,198),(17,234,199),(18,235,200),(19,236,201),(20,237,202),(21,238,203),(22,239,204),(23,240,205),(24,241,206),(25,242,207),(26,243,208),(27,244,209),(28,245,210),(29,246,106),(30,247,107),(31,248,108),(32,249,109),(33,250,110),(34,251,111),(35,252,112),(36,253,113),(37,254,114),(38,255,115),(39,256,116),(40,257,117),(41,258,118),(42,259,119),(43,260,120),(44,261,121),(45,262,122),(46,263,123),(47,264,124),(48,265,125),(49,266,126),(50,267,127),(51,268,128),(52,269,129),(53,270,130),(54,271,131),(55,272,132),(56,273,133),(57,274,134),(58,275,135),(59,276,136),(60,277,137),(61,278,138),(62,279,139),(63,280,140),(64,281,141),(65,282,142),(66,283,143),(67,284,144),(68,285,145),(69,286,146),(70,287,147),(71,288,148),(72,289,149),(73,290,150),(74,291,151),(75,292,152),(76,293,153),(77,294,154),(78,295,155),(79,296,156),(80,297,157),(81,298,158),(82,299,159),(83,300,160),(84,301,161),(85,302,162),(86,303,163),(87,304,164),(88,305,165),(89,306,166),(90,307,167),(91,308,168),(92,309,169),(93,310,170),(94,311,171),(95,312,172),(96,313,173),(97,314,174),(98,315,175),(99,211,176),(100,212,177),(101,213,178),(102,214,179),(103,215,180),(104,216,181),(105,217,182)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105),(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210),(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315)])

315 conjugacy classes

class 1 3A···3H5A5B5C5D7A···7F15A···15AF21A···21AV35A···35X105A···105GJ
order13···355557···715···1521···2135···35105···105
size11···111111···11···11···11···11···1

315 irreducible representations

dim11111111
type+
imageC1C3C5C7C15C21C35C105
kernelC3×C105C105C3×C21C3×C15C21C15C32C3
# reps1846324824192

Matrix representation of C3×C105 in GL2(𝔽211) generated by

1960
01
,
450
093
G:=sub<GL(2,GF(211))| [196,0,0,1],[45,0,0,93] >;

C3×C105 in GAP, Magma, Sage, TeX

C_3\times C_{105}
% in TeX

G:=Group("C3xC105");
// GroupNames label

G:=SmallGroup(315,4);
// by ID

G=gap.SmallGroup(315,4);
# by ID

G:=PCGroup([4,-3,-3,-5,-7]);
// Polycyclic

G:=Group<a,b|a^3=b^105=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C105 in TeX

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